Problem 7 - Applied Calculus

Question

A town has a population of 1000 people at time \(t=0\) In each of the following cases, write a formula for the population, \(P\), of the town as a function of year \(t\) (a) The population increases by 50 people a year. (b) The population increases by \(5 \%\) a year.

Step-by-Step Solution

Answer

(a) \( P(t) = 1000 + 50t \), (b) \( P(t) = 1000 \times (1.05)^t \)

Step 1: Identify Initial Conditions

The initial population at time \(t = 0\) is \(P(0) = 1000\). This serves as the starting point for our formula.

Step 2: Develop Formula for Part (a)

In Part (a), the population increases by a constant 50 people each year. Thus, the formula for the population as a function of time is: \[ P(t) = 1000 + 50t \] where \(t\) is the number of years since the initial time.

Step 3: Develop Formula for Part (b)

In Part (b), the population increases at a rate of \(5\%\) per year. This is an exponential growth problem. The formula for the population as a function of time is: \[ P(t) = 1000 \times (1 + 0.05)^t \] This formula accounts for the cumulative percentage increase over time.

Key Concepts

Linear Growth

Linear growth describes a situation where a quantity increases by a fixed amount each time period. In the context of our population example, linear growth is demonstrated by the town's population increasing by 50 people per year. This consistent rate of increase means that the change in population is directly proportional to time.

To express this mathematically, we use the formula:

• \( P(t) = P(0) + rt \)

where:

• \( P(t) \) is the population at time \( t \)
• \( P(0) \) is the initial population
• \( r \) is the constant rate of increase (50 people per year)
• \( t \) is time in years since the initial point

In our problem, this is clearly demonstrated by the formula \( P(t) = 1000 + 50t \). Each year the population grows by exactly 50 people, no more, no less, resulting in a straight-line graph when plotted over time.

Exponential Growth

Exponential growth occurs when a quantity increases by a consistent percentage rate over time. This type of growth is not linear; instead, it accelerates as time progresses, meaning the rate of increase is proportional to the current population. For our town, an annual growth of 5% means that each year the population increases by 5% of its current size, which is reflected in the formula:

• \( P(t) = P(0) \, (1 + r)^t \)

where:

• \( P(t) \) is the population at time \( t \)
• \( P(0) \) is the initial population
• \( r \) is the growth rate as a decimal (0.05 for 5%)
• \( t \) is time in years since the initial point

The solution presents this with \( P(t) = 1000 \times (1 + 0.05)^t \). This formula captures the compounding effect of exponential growth, leading to a curve that steepens over time, showing how the population grows more quickly each successive year.

Initial Conditions

Initial conditions are critical when setting up any population growth problem, whether it involves linear or exponential growth. They represent the starting point from which all calculations begin.

For our exercise, at time \( t=0 \), the population \( P(0) \) is 1000 people. This information establishes the baseline from which growth is measured.

Having a clear understanding of the initial conditions allows us to correctly formulate both linear and exponential growth models. In both types of growth, these initial conditions ensure that calculations are aligned with real-world observations right from the start.

• Initial conditions help us position the model accurately in reality.
• They provide context, allowing us to predict future values based on past data.
• They are essential for extrapolating trends in both types of growth scenarios.

Accurately identifying and using initial conditions provides the crucial starting point for crafting formulas that reflect the actual dynamics of the situation.